General Mathematics

2610 Submissions

[4] viXra:2610.0084 [pdf] submitted on 2026-10-23 04:44:01

Comparative Efficiency of Methods for Minimizing Implicit Unimodal Functions

Authors: Vladimir Kodnyanko
Comments: 14 Pages.

This paper examines combinations of methods for minimizing implicit unimodal functions. These include transformative methods, which convert implicit functions into explicit functions, and minimizing functions, which minimize the transformed functions. Transformative methods include bisection, regula falsi, and Ridders methods, as well as two modifications of the new optisection method and two modifications of parabolic bisection. The minimizing functions used are bisection, the golden ratio, and two variations of optisection. Examples of minimizing strictly unimodal and monotone implicit functions are given. For rapid minimization of the former, a combination of minimizing parabolic optisection and transformative combined bisection, which combines the advantages of conventional bisection, regula falsi, and parabolic methods, should be given. Minimizing monotone implicit functions should be minimized by minimizing optisection methods, which have the highest convergence rate. In this case, preference should be given to optisection and parabolic bisection methods as transformative methods. This choice allows the minimization of implicit unimodal functions to be performed ten or more times faster than other combinations of methods.
Category: General Mathematics

[3] viXra:2610.0042 [pdf] submitted on 2026-10-10 14:12:38

A Recreational Exploration of Near-Miss Constructions for a 3×3 Magic Square of Square Numbers: The Error Function, Eisenstein Triangles, and Continued Fractions as Tools for Investigating Progressively Better Approximations

Authors: Arina Bator
Comments: 21 Pages.

This work falls within the scope of recreational mathematics and the experimental exploration of the order-3 magic square of squares. Instead of resolving the open problem of the existence of such an object, the research focuses on "near-miss" structures. The analysis is based on the reduction of the original problem to three three-term arithmetic progressions of square numbers with a common difference (r), whose first terms also form an arithmetic progression.By relaxing the condition requiring the initial terms to form an arithmetic progression, this paper introduces a percentage error function (p). This function allows for a quantitative assessment of the discrepancy between the differences of these terms, and consequently, a quantitative measure of how closely a configuration approximates a fully magic structure.Subsequently, an algorithm developed for this study was applied to generate all three-term progressions of integer squares with a common difference r. To optimize calculations and eliminate redundant operations, a filter based on the theory of quadratic residues in the ring Z_240 was implemented. Ultimately, a deterministic exploration of the differences r from 2 to 70,000,000,000,000 was conducted.Next, a geometric parametrization utilizing 120-degree Eisenstein triangles is presented. This enabled a dimensionality reduction of the error optimization problem to a function of a single real variable. It was demonstrated that the theoretical optimum point with zero error corresponds to an irrational number, preventing its direct realization in integers. It was proven, however, that despite structural limitations, it is possible to construct a parametric matrix generating semi-magic squares with equal sums in rows, columns, and on one diagonal. Diophantine approximation and continued fractions yielded an infinite sequence of rational approximations where the error function converges asymptotically to zero, while the ratio of the diagonal sums remains close to 1. This is illustrated by a numerical example where the ratio yields a value on the order of 1 - 7.3506 x 10^(-103230).The results offer a new method for the assessment of near-miss structures based on the error function p. They outline an approach based on Eisenstein triangles, for generating configurations where the error p tends to zero and the diagonal sum ratio approaches 1. This work does not prove the non-existence of the magic square of squares, but provides a universal research framework for approaching ideal structures.
Category: General Mathematics

[2] viXra:2610.0034 [pdf] submitted on 2026-10-10 00:36:23

On the Uniqueness of Solutions to the Cauchy Problem for Systems of Linear Differential Equations with Partial Derivatives

Authors: Igor Shchitov
Comments: 6 Pages.

This article presents examples of linear partial differential equations with analytic coefficients and system of such equations for which the solutions to the Cauchy problem in the class of continuously differentiable functions are non-unique. More precisely, the time during which the analytical solution to the Cauchy problem remains unique is finite for such equations and system,after which this solution may even transform into one of many possible solutions. These solutions may have unusual properties.Key words: systems of linear partial differential equations; analytical coefficients; Cauchy problem; Holmgren’s theorem; uniqueness of solution.
Category: General Mathematics

[1] viXra:2610.0012 [pdf] submitted on 2026-10-03 20:31:44

DPSE: Difference Power Space Extrapolation for π Computation in O(√n) Iterations

Authors: Xingfeng Chen
Comments: 15 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We present Difference Power Space Extrapolation (DPSE), a geometric method for computing the n decimal digits of π by combining multiple polygon observations in difference power space. The paper shows that DPSE should be understood as a higher-order development of the Difference-Power Cumulative Method (DPCM), rather than asan unrelated construction. For the polygon semiperimeter surrogate Sk = nksin(π/nk), the geometric increments ∆Sk = Sk+1 − Sk admit a structured boundary-layer expansion that can be cancelled by Richardson-type linear combinations. For fixed extrapolation orderW, this yields the error law Ek,W = O(4−(W+1)k), while the diagonal Neville sequence exhibits an approximately quadratic error exponent in the physical observation index k. Consequently, the required physical observation count for n correct decimal digits is reduced from lineargrowth to order O(√n), while preserving the low-memory character of geometric iteration. Numerical experiments include a direct milliondigit fingerprint verification.
Category: General Mathematics